The Stacks project

Proof. Write $X = \overline{X} - \{ x_1, \ldots , x_ r\} $. Then $\mathop{\mathrm{Pic}}\nolimits (X) = \mathop{\mathrm{Pic}}\nolimits (\overline{X})/ R$, where $R$ is the subgroup generated by $\mathcal{O}_{\overline{X}}(x_ i)$, $1 \leq i \leq r$. Since $r \geq 1$, we see that $\mathop{\mathrm{Pic}}\nolimits ^0(\overline{X}) \to \mathop{\mathrm{Pic}}\nolimits (X)$ is surjective, hence $\mathop{\mathrm{Pic}}\nolimits (X)$ is divisible (see discussion in proof of Lemma 59.69.1). Applying the Kummer sequence, we get (1) and (3). For (2), recall that

where $\bar{\mathcal L} \in \mathop{\mathrm{Pic}}\nolimits ^0(\overline{X})$, $D$ is a divisor on $\overline{X}$ supported on $\left\{ x_1, \ldots , x_ r\right\} $ and $ \bar{\alpha }: \bar{\mathcal L}^{\otimes n} \cong \mathcal{O}_{\bar{X}}(D)$ is an isomorphism. Note that $D$ must have degree 0. Further $\tilde{R}$ is the subgroup of triples of the form $(\mathcal{O}_{\overline{X}}(D'), n D', 1^{\otimes n})$ where $D'$ is supported on $\left\{ x_1, \ldots , x_ r\right\} $ and has degree 0. Thus, we get an exact sequence

where the middle map sends the class of a triple $(\bar{ \mathcal L}, D, \bar\alpha )$ with $D = \sum _{i = 1}^ r a_ i (x_ i)$ to the $r$-tuple $(a_ i)_{i = 1}^ r$. It now suffices to use Lemma 59.69.1 to count ranks. $\square$